In the following our aim is to use our semi-parametric approach to construct a test whether the parametric fit suggested by [1]} is appropriate for the data. Let \(\varphi _{I,0}\)  and \(\varphi _{G,0}\)  denote the densities as defined in Section REF . More formally, we would like to test the hypotheses
\(H_0^{(I)}: \varphi _I=\varphi _{I,0}\quad \textrm {and}\quad H_0^{(G)}:\varphi _G=\varphi _{G,0}.\) 
